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Passive transport in the interstitium and circulation: basics
Published in Benjamin Loret, Fernando M. F. Simões, Biomechanical Aspects of Soft Tissues, 2017
Benjamin Loret, Fernando M. F. Simões
Passive transports are dissipative processes where the flux is directed against a certain gradient. By contrast, biology displays a number of examples where the flux takes place along the gradient. For example, cells move in directions where the density of adhesive sites is larger since they get a stronger grip to the matrix in these regions: the phenomenon is referred to as haptotaxis. Much in the same flavor, chemotaxis consists of the motion of cells towards a zone of higher nutrient content. Active transport across the corneal endothelium avoids swelling of the cornea and contributes to maintain its transparency. Active transports require energy. The topic is touched in Section 19.3.
Tumor Growth
Published in William E. Schiesser, Moving Boundary PDE Analysis, 2019
The physical interpretation of the various left and right hand side (LHS and RHS) terms in eq. (4.1) is discussed subsequently. Briefly, chemotaxis pertains to the movement of the cells in response to the gradient of the MDE. Haptotaxis pertains to the movement of the cells in response to a gradient in ECM.
Well-posedness of a mathematical model of diabetic atherosclerosis with advanced glycation end-products
Published in Applicable Analysis, 2022
We assume that all cells are moving with a common velocity [9,10]; the velocity is the result of movement of macrophages, T-cells and SMCs into the intima. We also assume that all species are diffusing with appropriate diffusion coefficients. The equation for each species of cells X has the form where the expression on the left side includes advection and diffusion, and accounts for various growth factors, bio-chemical reactions, chemotaxis and haptotaxis. The equation for the chemical species is the same but without the advection term. In this work, we consider only two-dimensional plaque for simplicity. Figure 2 shows a 2D cross-section of a blood vessel with plaque Ω.
Mechanobiological model to study the influence of screw design and surface treatment on osseointegration
Published in Computer Methods in Biomechanics and Biomedical Engineering, 2022
Nicolas Rousseau, Patrick Chabrand, Arnaud Destainville, Olivier Richart, Jean-Louis Milan
With and MSCs migration is considered to occur mainly through diffusion (Gruler and Bültmann 1984), chemotaxis (Lind et al. 1996) and haptotaxis (Carter 1965). These migrations are respectively driven by a diffusion function a haptotaxis one and a chemotaxis one While it was demonstrated that osteoblasts have the ability to migrate towards the implant surface (Jones and Boyde 1977), this migration is limited in comparison with MSCs as osteoblasts attached to the bone surface they secrete lose this ability (Davies 2003). Therefore, osteoblasts migration is neglected regarding MSCs one. In addition to migration, MSCs and osteoblasts proliferate respectively as a function of and that both also depend on mtot. MSCs differentiate depending on and osteoblasts apoptosis is considered by a simple linear relation, represented by a constant of decay Osteoblasts concentration then influence the secretion of woven bone at a rate depending on the functional form Qw. This woven bone subsequently remodels into lamellar bone at a rate that depends of Ql. Except for cells migration and apoptosis, all the aforementioned functional forms depend of the stimulus S and/or growth factors concentration. Finally, growth factors are considered to diffuse with a constant of diffusion Dg. They are produced as a function of osteoblasts concentration at a rate Eg and decay as a linear function of growth factors concentration at a rate dg. An overview of all variables, parameters, functions and associated references is available in the Appendix (Tables 1–3).
Qualitative behavior of solutions for a chemotaxis-haptotaxis system with gradient-dependent flux-limitation
Published in Applicable Analysis, 2022
In recent years, researchers have focused on mathematical models of cancer cell invasion. Chaplain and Lolas [22, 23] first proposed the chemotaxis-haptotaxis model, which describes cancer cells invasion into surrounding healthy tissue. More precisely, their model explains the chemotactic migration of tumor cells towards self-secreted diffusible chemicals and the haptotaxis migration towards the extracellular matrix (ECM). In fact, during the haptotaxis migration of cancer cell invasion, the ECM is degraded by matrix-degrading enzymes (MDE) that are produced by cancer cells, such as urokinase-type plasminogen activator (uPA) secreted by cancer cells, and this degradation allows cancer cells to migrate along the gradient of uPA (chemotaxis) in Ref. [24]. When cancer cells interact with the fibers of the extracellular matrix, the corresponding movement of the cancer cells responds to the gradient of non-diffusive extracellular matrix macromolecules, such as vimentin (haptotaxis) in Ref. [25]. Based on the model proposed in Refs [22, 23], Tao and Winkler [26] considered the following parabolic-elliptic-ODE system and they obtained the global boundedness and asymptotic stability of solutions for when . Similar results on the global boundedness and asymptotic behavior of solutions have been proved in Ref. [27] for . The results for the boundedness of the nonlinear diffusion of model (4) have been studied in Ref. [28]. For the fully parabolic-ODE system, similar results for the global boundedness of solutions have been proved for large μ in three dimensions (see Ref. [29]) and higher dimensions (see [30]), and asymptotic behavior of classical solution has been proved for in Ref. [31] and arbitrary in Ref. [32], and the boundedness of the associated gradient-dependent flux-limited model has been considered in Ref. [33]. When , the model (4) is simplified to the chemotaxis-only system. The main issue for the chemotaxis-only system is whether the solution of the system is bounded or blow-up, which can be found in many literature, such as Refs [4, 34–36] and references therein. When , the model (4) becomes a haptotaxis-only system, there exist some mathematical literature for haptotaxis-only models, please see Refs [37–43] and references therein.